1. Line a large baking sheet with parchment paper. Beat butter with a stand mixer or hand mixer on medium speed until light and creamy, 1 to 2 minutes. With mixer on low speed, gradually add powdered sugar, and beat until mixture forms a stiff, crumbly dough, 1 to 2 minutes, stopping to scrape down sides of bowl as needed. Add evaporated milk, peppermint extract, vanilla extract, and fine sea salt. (If using peppermint oil, start with 6 drops, and taste the dough to see if a stronger peppermint flavor is desired before adding more oil.) Beat on medium speed until mixture is smooth and soft and has the consistency of modeling clay, 2 to 3 minutes.
2. Remove dough from mixer bowl, and shape into a smooth ball. Cut ball in half; wrap 1 dough ball half in plastic wrap, and set aside. To color the remaining dough ball half, use a wooden pick to deposit a little of the red food coloring gel onto dough; massage in with your fingers until evenly blended. Keep adding food coloring gel, a little at a time, until it reaches the desired shade of red.
3. Divide red dough portion and reserved white dough portion into 4 equal portions each to make 8 equal portions (4 red and 4 white). On a flat work surface sprinkled with some powdered sugar, gently knead together 2 red portions of dough with 2 white portions of dough until the mixture looks marbled. (Do not overmix.) Roll into a long, smooth, 1⁄2-inch-thick log with visible swirls of both red and white color showing. Using a sharp knife, cut log into 60 (1⁄2-inch-thick) pieces. Repeat process with remaining 2 red dough portions and 2 white dough portions. Transfer buttermints to prepared baking sheet.
4. Let buttermints dry at room temperature, uncovered, 24 hours. (They’ll develop a crisp outer shell with a sweet, creamy, minty middle.) At this point, buttermints can be bagged up for giving as gifts, spooned into a candy dish, or stored in an airtight container or ziplock plastic bag for up to 2 weeks.